= Expected utility of a Gaussian location-scale family
For $X=\mu+\sigma Z$ with $Z$ having the <standard normal distribution> and an increasing concave <utility function> $U$, define $f(\mu,\sigma)=\mathbb E[U(X)]$. Then $f_\mu=\mathbb E[U'(X)]>0$ and <Gaussian integration by parts> gives $f_\sigma=\sigma\mathbb E[U''(X)]\leq0$. Moreover
$$
\operatorname{Hess}f
=\mathbb E\left[U''(X)
\begin{pmatrix}1&Z\\Z&Z^2\end{pmatrix}\right]
$$
is negative semidefinite, so $f$ is jointly <concave> in $\mu$ and $\sigma$.
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